Magnetohydrodynamic Motions of a Colloidal Sphere in a Concentric Spherical Cavity

• Main Authors: Tzu-Hsien Hsieh, 謝子賢
• Other Authors: 葛煥彰
• Format: Others
• Language: en_US
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/69394379710876101320

博士 === 國立臺灣大學 === 化學工程學研究所 === 101 === The quasi-steady motions of a spherical colloidal particle inside a concentric spherical cavity filled with a conducting fluid induced by the magnetohydrodynamic (MHD) effect are analyzed at low Reynolds number. Through the use of a generalized reciprocal theorem to the Stokes equations modified with the Lorentz force density resulting from the interaction of an applied magnetic field with the existing electric current and the consideration of the Maxwell stress to the force exerted on the particle, the translational and angular velocities of the particle under various conditions are obtained in closed forms valid for an arbitrary value of the particle-to-cavity radius ratio. The boundary effects on the motions of the particle caused by the MHD force are generally equivalent to (yet different from) that in sedimentation. In Chapter 2, an analytical study is presented for the MHD effects on a translating and rotating charged sphere in an arbitrary unbounded electrolyte solution prescribed with a general flow field and a uniform magnetic field. The electric double layer surrounding the charged particle may have an arbitrary thickness relative to the particle radius. Through the use of a simple perturbation method, the Stokes equations modified with an electric force term, including the Lorentz force contribution, are dealt with using a generalized reciprocal theorem. Using the equilibrium double-layer potential distribution from solving the linearized Poisson-Boltzmann equation, we obtain closed-form formulas for the translational and angular velocities of the spherical particle induced by the MHD effects to the leading order. It is found that the MHD effects on the particle movement associated with the translation and rotation of the particle and the ambient fluid are monotonically increasing functions of ka , where k is the Debye screening parameter and a is the particle radius. Any pure rotational Stokes flow of the electrolyte solution in the presence of the magnetic field exerts no MHD effect on the particle directly in the case of a very thick double layer. The MHD effect caused by the pure straining flow of the electrolyte solution can drive the particle to rotate, but it makes no contribution to the translation of the particle. In Chapter 3, the MHD effects on the translation and rotation of a charged sphere situated at the center of a charged spherical cavity filled with an arbitrary electrolyte solution when a constant magnetic field is imposed are analyzed. The electric double layers adjacent to the solid surfaces may have an arbitrary thickness relative to the particle and cavity radii. Through the same method of analysis in Chapter 2, we obtain explicit formulas for the translational and angular velocities of the colloidal sphere produced by the MHD effects valid for all values of the particle-to-cavity size ratio. The boundary effect on the MHD motion of the spherical particle is a qualitatively and quantitatively sensible function of the parameters a/b and ka , where b is the radius of the cavity. In general, the proximity of the cavity wall reduces the MHD migration but intensifies the MHD rotation of the particle. In Chapter 4, the electromagnetophoretic (EMP) motion of a spherical colloidal particle positioned at the center of a spherical cavity filled with a conducting fluid is analyzed. Under uniformly applied electric and magnetic fields, the electric current and magnetic flux density distributions are solved for the particle and fluid phases of arbitrary electric conductivities and magnetic permeabilities. Applying a generalized reciprocal theorem to the Stokes equations modified with the resulted Lorentz force density, we obtain a closed-form formula for the migration velocity of the particle valid for an arbitrary value of the particle-to-cavity radius ratio. The particle velocity in general decreases monotonically with an increase in this radius ratio, with an exception for the case of a particle with high electric conductivity and low magnetic permeability relative to the suspending fluid. The asymptotic behaviors of the boundary effect on the EMP force and mobility of the confined particle at small and large radius ratios are discussed. In Chapter 5, an analytical study is presented for the magnetic-field-induced motion of a colloidal sphere with spontaneous electrochemical reactions on its surface situated at the center of a spherical cavity filled with an electrolyte solution. The zeta potential associated with the particle surface may have an arbitrary distribution, whereas the electric double layers adjoining the particle and cavity surfaces are taken to be thin relative to the particle size and the spacing between the solid surfaces. The electric current and magnetic flux density distributions are solved for the particle and fluid phases of arbitrary electric conductivities and magnetic permeabilities. Applying a generalized reciprocal theorem to the Stokes equations with the resulted Lorentz force term, we obtain explicit formulas for the translational and angular velocities of the colloidal sphere valid for all values of the particle-to-cavity size ratio. The dipole and quadrupole moments of the zeta potential distribution over the particle surface cause the particle translation and rotation, respectively. The induced velocities of the particle are unexpectedly significant, and their dependence on the characteristics of the particle-fluid system is physically different from that for EMP particles or phoretic swimmers. The particle velocities decrease monotonically with an increase in the particle-to-cavity size ratio. The boundary effect on the movement of the particle with interfacial self-electrochemical reactions induced by the MHD force is much stronger than that in phoretic swimming.